A note on submanifolds and mappings in generalized complex geometry

TL;DR

This paper explores the properties of submanifolds and mappings within generalized complex geometry, providing criteria for smoothness, dual results for submersions, and insights into generalized Kähler submanifolds.

Contribution

It introduces a new expression for the induced generalized complex structure on submanifolds and analyzes invariant submanifolds in generalized Kähler manifolds.

Findings

Derived a smoothness criterion for submanifolds with induced structures

Dualized results to submersions in generalized complex geometry

Identified invariant submanifolds in generalized Kähler manifolds

Abstract

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.